Question: Rewrite the function by completing the square. $f(x)= x^{2} + x -30$ $f(x)=$
Answer: $\begin{aligned} f(x)&= x^2 + x -30 \\\\ &= \left(x^2 +x\right) -30 \end{aligned}$ Now we want to complete $x^2 + x$ into a perfect square. To do that, we should add $\left(\dfrac{{+1}}{2}\right)^2={\dfrac{1}{4}}$ to it: $x^2{+}x+{\dfrac{1}{4}}=\left(x +\dfrac{1}{2}\right)^2$ We add ${\dfrac{1}{4}}$ inside the parentheses, and subtract ${1}\cdot{\dfrac{1}{4}}$ outside them, to keep the expression equivalent. $\begin{aligned} &\phantom{=} \left(x^2 + x\right) -30 \\\\ &=\left(x^2 +x+{\dfrac{1}{4}}\right) -30 -{1}\cdot{\dfrac{1}{4}} \\\\ &= \left(x +\dfrac{1}{2}\right)^2 -30 -\dfrac{1}{4} \\\\ &= \left(x +\dfrac{1}{2}\right)^2 -\dfrac{121}{4} \end{aligned}$ In conclusion, the function after completing the square is written as: $f(x)= \left(x +\dfrac{1}{2}\right)^2 -\dfrac{121}{4}$